On variable step highly stable 4-stage Hermite-Birkhoff solvers for stiff ODEs

Variable-step (VS) \(4\)-stage \(k\)-step Hermite--Birkhoff (HB) methods of order \(p=(k+1)\), denoted by HB\((p)\), are constructed as a combination of linear \(k\)-step methods of order \((p-2)\) and a two-step diagonally implicit \(4\)-stage Runge--Kutta method of order 3 (TSDIRK3) for solving stiff ordinary differential equations. The main reason for considering this class of formulae is to obtain a set of \(k\)-step methods which are highly stable and are suitable for the integration of stiff differential systems whose Jacobians have some large eigenvalues lying close to the imaginary axis. The approach, described in the present paper, allows us to develop \(L\)-stable \(k\)-step methods of order up to 7 and \(L(\alpha)\)-stable methods of order up to 10 with \(\alpha > 64^\circ\). Fast algorithms are developed for solving confluent Vandermonde-type systems of the new methods in O\((p^2)\) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The step sizes of these methods are controlled by a local error estimator. Selected HB(\(p\)) of order \(p\), \(p=4,5,\ldots,9\), compare favorably with existing Cash modified extended backward differentiation formulae, MEBDF(\(p\)), \(p=4,5,\ldots,8\) in solving problems often used to test highly stable stiff ODE solvers on the basis of CPU time, number of steps and error at the endpoint of the integration interval.